Decay of the probability of ruin under uncertain investments

The classical result of the Cramer-Lundberg risk model states that if the rate of premium, c, exceeds the average of the claims paid per unit time, µ , then the probability of ruin of the insurance company decays exponentially fast as the initial capital u ! 1. This asymptotic behavior of the probability of ruin is derived again by means of infinitesimal gener- ators and Laplace transforms. Using the same tools, it is shown that the probability of ruin has an algebraic decay rate if the insurance company invests its capital in a risky asset with a price which follows a geometric Brownian motion. The latter result is shown to be valid not only for ex- ponentially distributed claim amounts, as in Frolova et al. (2002), but, more generally, for any claim amount distribution that has a moment gen- erating function defined in a neighborhood of the origin.